Publication: Sürekli fonksiyon halkalarının bazı özel idealleri
Abstract
SÜREKLİ FONKSİYON HALKALARININ BAZI ÖZEL İDEALLERİ Bu çalışmada herhangi bir topolojik uzayı üzerindeki sürekli fonksiyonlar halkası (veya ) in -idealleri ile in -süzgeçleri arasındaki ilişkiler ortaya konulmuş ve in asal ideallerinin dağılımı incelenmiştir. Ayrıca in asal ideali olmak üzere bölüm halkasının asalımsı idealleri incelenmiştir. Birinci bölümün ilk kısmında halka teoriyle ilgili temel tanımlar ve daha sonraki bölümlerde kullanılacak bazı teoremler verilmiştir. İkinci ve daha sonraki kısımlarda ise halkasının cebirsel yapısı ile sıralama yapısı incelenmiştir. İkinci bölümde in topolojik özellikleriyle in cebirsel özellikleri arasındaki ilişkiler incelenmiştir. Ayrıca, asal -süzgeçlerin bir genelleştirmesi olan 2-yutan -süzgeç tanımı verilmiş olup 2-yutan -süzgeçlerle 2-yutan -idealler arasındaki ilişkiler ortaya konulmuştur. Tezin üçüncü bölümünde nin sıralama yapısı ve deki asal ideallerin dağılımı incelenmiştir. Ayrıca, in bir asalını içeren asal ideallerin bir zincir oluşturduğu gösterilmiş ve asal -ideallerle ilgili bazı teoremler verilmiştir. Son bölümde tam sıralı halkasının asalımsı idealleri karakterize edilmiştir ve değişmeli bir halkada bir idealin asalımsı ayrışıma sahip olamayabileceği gösterilmiştir.
SOME SPECIAL IDEALS IN RINGS OF CONTINUOUS FUNCTIONS In this study, the relations between -ideals and -filters of rings of continuous functions (or ) on any topological space are given and also the distribution of prime ideals in is investigated. Furthermore, Let be a prime ideal in , then the primary ideals in residue class ring are studied. Therefore, Firstly in the first section of the first chapter of this thesis, fundemantal definitions and some theorems that will be used in other chapters are given. Also in the second and other parts of the first chapter, algebraic and order structure of are studied. In the second chapter, the relations between topological properties of and algebraic properties of are given. Moreover, the notion of 2-absorbing -filter which is a generalization of prime -filter is introduced and the relation between 2-absorbing -ideal of and 2-absorbing -filter on is given. In the third chapter, order structure of residue class ring and the distribution of prime ideals are investigated. Furthermore, it is shown that prime ideals containing a prime ideal in form a chain and some theorems about prime -ideals in are given. Finally, primary ideals of residue class ring are charectarized and also it is shown that an ideal may not be the intersection of primary ideals in a general commutative ring.
SOME SPECIAL IDEALS IN RINGS OF CONTINUOUS FUNCTIONS In this study, the relations between -ideals and -filters of rings of continuous functions (or ) on any topological space are given and also the distribution of prime ideals in is investigated. Furthermore, Let be a prime ideal in , then the primary ideals in residue class ring are studied. Therefore, Firstly in the first section of the first chapter of this thesis, fundemantal definitions and some theorems that will be used in other chapters are given. Also in the second and other parts of the first chapter, algebraic and order structure of are studied. In the second chapter, the relations between topological properties of and algebraic properties of are given. Moreover, the notion of 2-absorbing -filter which is a generalization of prime -filter is introduced and the relation between 2-absorbing -ideal of and 2-absorbing -filter on is given. In the third chapter, order structure of residue class ring and the distribution of prime ideals are investigated. Furthermore, it is shown that prime ideals containing a prime ideal in form a chain and some theorems about prime -ideals in are given. Finally, primary ideals of residue class ring are charectarized and also it is shown that an ideal may not be the intersection of primary ideals in a general commutative ring.